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### Solution of the Relativistic Hartree-Fock Equations

#### 1966

WE give here a brief report on a computer program which is currently being constructed to solve the relativistic Hartree-Fock equations for an atom or ion of high atomic number. The basic theory has been known for some years [1, 2], as also the basic technique for solving the equations [3]. However the only solutions published to date have been for the simpler Hartree equations in which exchange is neglected [3, 4, 5]. The main reason for this is the large amount of storage required, and the large amount of machine time needed to generate solutions on slow and small machines.

The basic numerical problem is that of solving a set of coupled ordinary differential equations of the form

for i=1, 2, ..., N

These equations are written in atomic units in which c, the velocity of light, has the numerical value 137, approximately. The subscript i refers to the i-th one-electron orbital. The constant ki is related to the angular momentum associated with the i-th orbital, and Yi, a functional of the set of functions {Pj,Qj }, can be regarded as an effective nuclear charge associated with the i-th orbital. With the functionals WPi andWQi , which are similarly defined, Yi serves to couple the equations. The functions Pi and Qi represent the radial part of the relativistic spinor wave-function and (Pi 2 + Qi 2) is the charge density associated with the i-th orbital, normalized so that

In the non-relativistic limit ( c -> ∞), Pi becomes the usual wave-function and Qi vanishes. Boundary conditions are given for each pair of functicions (Pi, Qi) at r = 0 and as r -> ∞, and the number of zeros which the functions may have is also prescribed. We require the solutions (Pi, Qi) and the corresponding eigenvalues εi, i = 1, 2, ..., N, for which equations (1) are "self-consistent". An iterative process for securing self-consistency is given in reference [3] .

We can now see the reasons why so little work has been done to solve the relativistic Hartree- Fock equations even though the solutions would be extremely useful in other atomic calculations. Take, for example, the case of neutral mercury (atomic number 80), which has been studied by Mayers [3]. Here, the number of orbitals, N, is 22. Taking, say, 251 radial points - more may be needed - the storage required for the solutions alone is 2 × 22 × 250, or, say, of the order of 10K. The Y's and W's need only be computed as needed, and add about another 1K of storage. When storage for the program is taken into account, we see that at least 12-13K of fast store is needed for an efficient solution. Mayers (circa 1958) was able to use an IBM 704 with 32K of fast store, but his attempt to solve the Hartree-Fock equations took of order 3 hours. Some doubts remain about the accuracy of the Hartree-Fock solutions that he obtained, and he has never published them. The increased speed available on Atlas should bring the running time down to order 20 minutes, which is not unreasonable.

Most of the code has been written, and we hope to complete it in the course of the next few weeks. The work is being carried out in collaboration with D. F. Mayers of the Oxford University Computing Laboratory.

#### References

1. B. Swirles, Proc. Roy. Soc. 152, 635, 1935; corrections are given by D. R. Hartree, Reports on Progress in Physics 11, 113, 1948.

2. 1. P. Grant, Proc. Roy. Soc. A262, 555, 1961; and Proc. Phys. Soc. 86, 523, 1965: J.R.A. Cooper, Proc. Phys. Soc. 86, 529, 1965.

3. D. F. Mayers, Proc. Roy. Soc. A241, 93, 1957; unpublished Ph. D. thesis (Cambridge University).

4. A.C. Williams, Phys. Rev. 58, 723, 1940.

5. S. Cohen, Phys. Rev. 118, 489, 1960; Rand Corp. report RM-2272-AEC.